Development of a three-dimensional mixed variational model for woven composites. II. Numerical solution and validation

Abstract

A mixed three-dimensional variational model, derived in an adjoining paper, is solved numerically for stress analysis with a finite element approach. Since the mixed model calculates the stress field by taking variations of displacement and stresses independently and satisfying equilibrium of stresses pointwise, accurate interlaminar stresses are predicted at the yarn interface. The interface continuity conditions are implemented through a penalty method by adding an additional variational energy of two constraint conditions: the displacements must be continuous along the interface between two stacked subregions, and interfacial normal and shear stresses must be in equilibrium at the interface. After performing the thickness integration, the three-dimensional variational energy equation is evaluated for each yarn (subregion) two-dimensionally with 16 stress-related and 13 displacement-related unknown variables. Rayleigh–Ritz approximation yields a system of linear equations by taking derivatives of the variational energy equation with respect to the independent unknown variables. The present mixed method is applied to analyze a flat laminated composite with a free edge, and the representative volume element of woven fabric composites. The displacement and stress results of the present method are compared and validated with the conventional displacement-based finite element solutions and/or the previous analytic solution.